Both linear and angular metrics are important to analyses of head
injury mechanics. Linear and angular accelerations have been correlated
with brain injury, and linear acceleration has been correlated with
skull fracture [1, 2, 3, 4, 5, 6]. Many novel head injury metrics, such
as BrIC, GAMBIT, HIP, PRHIC, Principle Component Score (PCS), RIC, RVCI,
and [DELTA][[omega].sub.peak] require knowledge of angular terms
(velocities and accelerations) [7-8]. Angular terms are also influential
in determining linear acceleration injury metrics; as
six-degree-of-freedom (6DOF) data is required to transfer data collected
on the perimeter of an object (usually the head) to center of mass (COM)
linear accelerations [9, 10, 11]. 6DOF data is most important when
studying brain injury mechanisms and human tolerance to acceleration in
situations with complicated kinematics where the activity under study is
not amenable to high-speed video analysis [5, 7], such as helmet
impacts.

To understand brain injury tolerances, analysis of dynamic loadings
in real-world and testing environments must be investigated. Studies
performed with human subjects often use angular rate sensors as part of
6DOF systems due to their smaller geometric size requirement, reduced
channel count, and stability in long-duration measurements [12, 13, 14,
15, 16, 17, 18, 19]. It is increasingly common for researchers to
perform ATD testing with angular rate sensors mounted in a headform
either in replacement of a linear accelerometer array [20-21], or
supplementing one [22, 23, 24, 25]. However, while data processing and
filtering methods are standardized and straightforward when using a
linear accelerometer array [26], how to handle angular rate sensor data
is less established, especially for impact or other short-duration
events [26-27].

Measuring Angular Velocity and Acceleration

There are a variety of ways to either measure or calculate angular
velocity and angular acceleration. Theoretically, only six linear
accelerometers are needed to solve the three linear and three angular
terms, although in practice nine accelerometers are required because it
provides improved error-correction and an algebraic means of calculating
angular acceleration [27-28]. Two-dimensional (planar) solutions based
on linear accelerometers have existed since the late 1960s and the
three-dimensional solution since 1975 (Figure 1). also known as the
nine-accelerometer-package (NAP) [10].

3-2-2-2 Linear Accelerometer Array (NAP)

Padgaonkar et al. described a nine-accelerometer array of linear
accelerometers in a 3-2-2-2 configuration that offered matched pairs of
accelerometers to cancel Coriolis components of angular accelerations
[28]. This yielded angular acceleration Eqs (1-1) independent of angular
velocity terms. In these equations, A represents linear acceleration,
[rho] represents the offset distance from the center of mass, and
[omega] represents angular velocity. The subscripts describe the axis
and sensor position. Angular velocities were calculated as the numerical
time integral of the angular accelerometer data. DiMasi provided an
exhaustive analysis of the NAP implementation in standard Hybrid-III
headforms, including how to handle non-coplanar sensors, alignment
errors, and non-centroidal seismic mass locations for "COM"
accelerometers [10].

[mathematical expression not reproducible] (1)

[mathematical expression not reproducible] (2)

[mathematical expression not reproducible] (3)

The advantages of the 3-2-2-2 configuration were numerous versus
the six-channel configurations, at the cost of increased channel count
and size. The NAP configuration, due to the required numerical
integration step to calculate angular velocity, is prone to cumulative
integration error from bias, drift, misalignment, or vibration. Any
measurement error will be propagated through all later times [27, 29].

The 3-2-1 or 3-1-1-1 Array

Although theoretically equivalent to the 3-2-1 configuration, in
practice the 3-1-1-1 is not favored, as it offers no advantages over the
3-2-1 design in terms of channel count or mathematical rigor, but takes
up more space by its three-dimensional nature. The 3-2-1 array is
generally presented as a configuration consisting of a triaxial linear
acceleration unit at the origin with a pair of y-axis and z-axis
oriented accelerometers on an arm along the x-axis with a single z-axis
oriented accelerometer located on an arm along the y-axis; the equations
for angular acceleration for this configuration consist of Eqs 4-6 [27,
30, 31, 32].

[mathematical expression not reproducible] (4)

[mathematical expression not reproducible] (5)

[mathematical expression not reproducible] (6)

Eqs 4-6 demonstrate that the angular acceleration components
require knowledge of the angular velocity terms. Although these
differential equations can be solved numerically, the solution is
unstable and because of errors in the experimentally measured
accelerations, stepwise integration results in an accumulation of error
in the angular accelerations that in turn propagate to the angular
velocity terms [31-32]. The result is angular terms that rapidly lose
accuracy, regardless of the how the six linear accelerometer channels
are analyzed [27, 30, 31, 32].

Mixed Arrays of Linear Accelerometers and Angular Rate Sensors

Studies have also used arrays with more exotic combinations of
linear accelerometers [33-34] or 6DOF systems using combinations of
triaxial linear accelerometers and triaxial angular rate sensors. Such
arrays have been favorably compared to the angular velocity and angular
acceleration calculations of linear accelerometer arrays [23, 35, 36,
37]. They offer direct measurement of angular velocity terms and the
advantage of reduced size and channel count, which is useful for studies
involving sensors worn for more than a brief period of time. The primary
difficulty with the use of angular rate sensors is the numerical
differentiation required to calculate angular accelerations. Noise
components are preferentially increased by numerical differentiation,
and over-filtering is usually employed to compensate for this [38]. The
error caused by differentiation is transient and applies only to each
individual time step, as opposed to the persistent nature of integration
errors.

The 3a[omega] Array

An increasingly common mixed array is a combination of three linear
accelerometers and three angular rate sensors, often implemented as two
triaxial arrays or one cluster with the three angular rate sensor
channels mounted to their corresponding channel of a triaxial linear
accelerometer. This array type has been shown to be effective for
determining angular kinematics, including during impacts [35, 36,
37,39]. A six-channel (3a[omega]) combination of linear accelerometers
and angular rate sensors is theoretically advantageous in that no such
calculations are needed to directly measure six terms, and angular
accelerations can be independently calculated as the time derivatives of
the angular velocity terms [24, 27, 36, 40]. Because multiple sensor
locations are not required, direct measurement of angular velocities is
less susceptible to error from bias, drift, and misalignment.

The primary limitation of 3a[omega]-based arrays is that a
numerical differentiation step is required to calculate the angular
accelerations.

Theoretically, differentiation error does not accumulate over time
as integration error does, and is less sensitive to drift and bias [27].
However, differentiation preferentially amplifies high-frequency noise
and can lead to grossly inaccurate angular acceleration results [24, 27,
38]. In practice, numerical differentiation of inherently noisy
experimental data is problematic, and requires care in the selection of
filtering methods. Unfortunately, there is little standardization in how
data should be filtered prior to differentiation, and methods in the
literature vary considerably [13, 14, 19, 21, 36, 40].

The 6a[omega] Array

One means of avoiding both the numerical integration step required
by the 3-2-2-2 linear array and the numerical differentiation step
required by the 3a[omega] array is to construct a 6a[omega] array, which
can take the form of a 3-2-1 array or a 3-1-1-1 array with an additional
triaxial array of angular rate sensors. Kang et al. provided an analysis
of a 3-1-1-1 array with the angular rate sensors mounted to each
non-centroidal arm [25]. They found that the 6a[omega] array yielded
angular acceleration results that were closer to that of the NAP than a
3a[omega] array, transformed peripheral linear accelerations to the COM
better than the 3a[omega] array and at least as well as the NAP, yielded
angular velocity data more accurate than the NAP, and provided angular
velocity data which could be integrated into angular displacement data
at higher accuracy than the NAP [24-25]. With sufficiently capable
angular rate sensors, all channels of the 6a[omega] array can be
filtered to CFC1000, which preserves time-integrity for transformation
purposes [24].

The limitation of 6a[omega]-based arrays is that they are just as
channel count and geometrically demanding as the 3-2-2-2 NAP, and the
individual sensors are often slightly heavier than the linear
accelerometers they replace. This can make the 6aco array difficult to
accommodate in applications with limited space, and it is a more costly
solution.

Angular Accelerometers

There is a paucity of data on the use of sensors which directly
measure angular acceleration of the head. Angular accelerometers have
mostly been used for modal analyses [42], and little is available
regarding their efficacy in an automotive or biomechanical context.
These sensors tend to be larger and with a lower frequency response than
the linear accelerometers and angular rate sensors typically used. A
CFC1000-capable angular accelerometer does exist (K-Shear[R] 8838/8840,
Kistler Instrument Corporation, Amherst, NY), however it is
approximately four times larger and six times more massive on a
per-channel basis than a comparable angular rate sensor. A 3-3 array of
linear and angular accelerometers (3a[alpha]) may offer no advantages
over a 3-3 array of linear accelerometers and angular rate sensors
(3a[omega]), because it merely trades the problems of integration for
differentiation, and the time-history for integration accumulates error.
A 3-3-3 array of linear accelerometers, angular rate sensors, and
angular accelerometers is viable and would directly measure all nine
quantities. However, this array offers no means of redundantly
calculating linear accelerations, unlike the 3-2-2-2 or 6a[omega]
arrays.

Filtering

For short-duration impact analyses, the numerical integration
required to calculate angular velocity by arrays of linear
accelerometers is generally preferred to the numerical differentiation
required to calculate angular accelerations from angular velocities, as
numerical integration acts as a low-pass filter to small errors in
angular acceleration calculations. However, these integrations are
stable for only a short period of time (on the order of seconds), before
cumulative error from sources such as inaccurate zero-bias, sensor
drift, and small mismatches between sets of accelerometers overwhelm the
signal under analysis [27, 28, 29, 30, 32]. This is an obstacle to
impact measurement in scenarios where there may be a series of discrete
impacts, where there is significant non-negligible pre-impact motion, or
where initial angular velocity is non-zero and of relevance. Examples of
these scenarios would be occupant kinematics in rollovers,
bicycle-automobile collisions, or forklift tipover or off-dock
scenarios.

Short-duration (0-200 ms) Events

For short-duration events, linear arrays of accelerometers are
customary and have been in wide use since the mid-1970s. Filtering for
linear accelerometers is straightforward and defined by SAE J211 as a
CFC1000 (1650 Hz cutoff) low-pass filter for head accelerations [26].
The state of the art is less defined for angular rate sensors. SAE J211
specifies no appropriate channel frequency class for differentiation of
angular rate sensor data, and the footnote referenced defines a means of
analyzing long-duration events [27].

Many authors have analyzed power spectral densities on a
case-by-case basis, or simply resorted to an ad hoc cutoff of 300 Hz
(sometimes described as CFC 180) [13, 14, 19, 21, 24]. Occasionally,
higher cutoff frequencies are chosen [36 40-41]. Laughlin compared the
angular accelerations from differentiated 1000 Hz magnetohydrodynamic
(MHD) angular velocity data to calculated angular accelerations from a
3-2-2-2 linear accelerometer array and found the MHD data to be equal to
or superior to the linear array data [40]. Martin et al. investigated
MHD angular rate sensors versus a rotary potentiometer and 10000 fps
high-speed film, and found 600 Hz to be an appropriate low-pass filter
cutoff [36]. Marshall and Guenther compared multiple types of rate
sensors to potentiometer data, a 3-2-2-2 linear accelerometer array, and
high-speed video data, and found MHDs had acceptable results at CFC1000
for pendulum head-form drops with and without impact [35]. It is
noteworthy that the ARS-1 MHD used in that study only had a frequency
response up to 1000 Hz; a 1650 Hz low-pass filter would have been of
limited use to that rate sensor. Voo et al. similarly compared an ATA
DynaCube 3 MHD to high-speed video and a 3-2-2-2 linear accelerometer
array, with all electronic data filtered at CFC1000. However, the
authors found that the angular velocity data was limited to 100 Hz and
the linear acceleration data was limited to 300 Hz, so lower cutoff
frequencies could have been used [37], Kang et al. in an analysis of the
6a[omega] array, also processed the data as a 3a[omega] array. They
found using low-pass filters of CFC60 (100 Hz) appropriate for
low-severity impacts and CFC180 (300 Hz) appropriate for high-severity
tests. CFC60 was found to overly reduce peak angular acceleration for
high-severity impacts. In general, they found that the 3a[omega] array
was either excessively noisy or excessively reduced in peak magnitude
[24]. Funk et al., in a mouthpiece-based study, low-pass filtered all
head kinematic data at CFC180. For differentiation into angular
acceleration data, they filtered both prior to and after numerical
differentiation [14], Camarillo et al. in a comparison to a mouthpiece
accelerometer system, low-pass filtered both angular rate sensor and
linear accelerometer data to CFC180 prior to transformation [13].
Suderman et al. analyzed impacts to an ATD wearing a hardhat, and, based
on a power spectral density analysis, determined that a 100 Hz low-pass
filter was appropriate for all tests in that series [21]. Lloyd and
Conidi indicated they filtered COM linear accelerations to CFC1000 and
COM angular rate sensor data via a phaseless 8th-order Butterworth
low-pass filter with a 500 Hz cutoff. The data were numerically
differentiated using a 5-point least-squares quartic equation [41]. For
purposes of comparison to the NAP, Siegmund et al., filtered mouthpiece
angular rate sensor data to 300 Hz, using a phaseless 4-pole Butterworth
filter [19].

Long-Duration (>200 ms) Events

Bussone et al. found per-event appropriate low-pass filter cutoff
frequencies for differentiating the angular rate sensor data into
angular accelerations by using a residual analysis to find individual
cutoff frequencies for the three center of mass (COM) linear
accelerometer channels and the three angular rate sensor channels and
taking the arithmetic mean of the six cutoffs. This was effective
because residual curves were reasonably insensitive to small variations
in the cutoff frequency in the band between linear and rotational cutoff
frequencies. The angular rate sensors provided more accurate rotational
velocities than integrated angular accelerations calculated from a
3-2-2-2 NAP, especially for events lasting longer than 200 ms [27].

Optimal Filtering

Filtering is a process that seeks to balance signal preservation
and noise reduction. Any analysis of a real signal requires a compromise
between the amount of distortion of true signal accepted and the amount
of noise rejected. By necessity, as one reduces noise passed one also
increases signal lost. In theory, a given angular velocity signal
contains a maximum signal frequency and an optimal cutoff frequency
which yields a subjectively noise-free velocity and acceleration which
would allow a "true differentiation" of the whole signal [38].
A method used in kinematics analyses and employed in prior analyses of
differentiation of angular velocities of the head is to self-analyze a
given channel's frequency content and accept the point at which the
lost signal and lost noise are equal, or an analysis of residuals [43].
Winter's method quantifies the difference between the filtered and
unfiltered signals as a function of the filter cutoff frequency across a
wide range of possible filter frequencies. The calculation of residuals
is shown in Eq 7.

[mathematical expression not reproducible] (7)

where [X.sub.i] is raw data at the [i.sup.th] sample and [X.sub.i]
is filtered data at the [i.sup.th] sample.

A theoretical signal containing only random noise would exhibit a
residual plot consisting of a straight line decreasing from an intercept
at 0 Hz to an intercept on the abscissa at the Nyquist frequency (half
the sample rate). The magnitude of the intercept value would represent
the root-mean-square (rms) value of the noise. When a signal consists of
a true signal combined with noise, the residual will rise above the
straight line as the cutoff frequency is reduced. This represents the
signal distortion caused as the cutoff is reduced. By setting equal the
amount of signal distorted with the amount of noise passed, a cutoff
frequency can be chosen. This is found by projecting a horizontal line
from the noise intercept until it intercepts the residual line. The
index of the intersection is an optimal cutoff frequency. This is
depicted in Figure 2. This method assumes noise is randomly distributed
and white, and the true signal of interest has a maximum frequency below
the Nyquist frequency. This method has been found to be effective for
long-duration head accelerations generated by inertial head motions [27]
because it tends to arrive at a solution range consistent with
least-squares matches to reference solutions. It has not yet been
validated for impact-based head kinematics.

A complication of filtering three-dimensional kinematics data is
that all three axes channels must be filtered to the same cutoff
frequency. Because low-pass filters shift data in the time domain as a
function of order and frequency, to avoid inconsistent time-shifting of
filtered data, all axes must be filtered to the same cutoff.
Accordingly, choosing a universal cutoff requires balancing the
filtering needs of each component channel. This is also a complication
of integrating angular rate sensor data with linear accelerometer data,
especially when the frequencies responses and filtering requirements are
not equal.

Arrays of linear accelerometers, specifically the 3-2-2-2
configuration, are time-proven, standards-compliant, and validated for
analysis of impact events. However, it is increasingly common for
researchers to use angular rate sensors to capture 6DOF kinematics at
reduced cost or with reduced channel counts. Less settled is an
equivalent appropriate filtration for angular rate sensor data into
angular acceleration data by numerical differentiation for impact
events. The purpose of this study was to identify an SAE J211-compliant,
a priori filtering method for angular velocity data in short-duration
(impact) events that most closely matched angular acceleration data
produced by linear arrays of accelerometers, and then to generalize that
method to determine appropriate filters for angular rate sensor data in
impact events. This was performed via a combination of physical testing
and multibody simulation. Physical testing provided performance data in
real-world environment that allowed the analyses of rate sensor data to
be compared to an array of linear accelerometers. The multibody
simulation provided data and kinematic results which were objectively
correct and not dependent on an intermediate comparison to a reference
standard, which while validated and widely-accepted, is not without
limitation.

METHODS

ATD Testing

The head of a 50th percentile male Hybrid-Ill ATD (1846-D head,
Denton ATD, Milan, Ohio) was instrumented with nine linear
accelerometers (7264-2000, Endevco, San Juan Capistrano, California) in
a 3-2-2-2 configuration in accordance with DiMasi [10]. The full scale
range of the accelerometers was +/- 2000 Gs with a resolution of 0.061
Gs/bit (16-bit system). The ATD head was also instrumented with triaxial
angular rate sensors (ARS-1500, Diversified Technical Systems, Seal
Beach, California). Channel orientations were in accordance with the SAE
J211-2014 standardized dummy head coordinate system: x-axis positive
forward, y-axis positive right, z-axis positive down [26]. The full
scale range of the angular rate sensors was +/- 1500 deg/sec, with a
resolution of 0.732 deg/sec/bit (12-bit system). To record the data, the
linear accelerometers were connected to a 16-bit data acquisition system
(PicoDAS, EME Corporation, Arnold, Maryland). The rate sensors were
connected to a 12-bit data acquisition system (NanoDAS, EME Corporation,
Arnold, Maryland), as less resolution was necessary for the rate sensors
due to the acceleration events using more of their dynamic range. The
data acquisition systems were sampled at frequencies up to 20 kHz per
channel.

The ATD was exposed to direct impact head motions with a rigid
rubber-covered mallet about each axis for linear and angular motion by
the following mechanisms:

1. The ATD was struck in the face at approximately the level of the
COM.

2. The ATD was struck in the head from the right side at
approximately the level of the COM.

3. The ATD was struck in the head from the right side more
forcefully at approximately the level of the COM.

4. The ATD was struck near the vertex of the head from the right
side.

5. The ATD was struck upwards at the chin.

6. The ATD was struck in the face from the right side at the level
of the nose.

All ATD data processing was completed using GNU Octave (4.0.1).
Digital filtering was done using phaseless, four-pole Butterworth
low-pass filters, in accordance with SAE J211-2014 [26, 44-45]. Data
were untrimmed and feature separate impacts with interval periods
without motion. Linear accelerometer data were filtered to CFC600 to
match the frequency response characteristics of the ARS-1500 rate
sensor, in order to make a like-for-like comparison.

Differentiation of filtered angular velocity data into angular
acceleration data was performed using three different numeric
differentiation equations:

[mathematical expression not reproducible] (8)

[mathematical expression not reproducible] (9)

[mathematical expression not reproducible] (10)

In these equations, f represents the value of the angular velocity
data at a specific point in time, x represents the specific point in
time, and h represents the difference in quantity between two adjacent
points in time.

Angular rate sensor data was analyzed by sweeping cutoff
frequencies from 1-1000 Hz to determine a single cutoff frequency that
best matched the resultant on a sum-of-the-square basis and which best
matched the peak value of the resultant. Data were also analyzed to
determine cutoff frequencies that yielded least-square or peak value
matches within [+ or -]5% of the optimum value.

Data from the angular rate sensors were validated against the
3-2-2-2 NAP results. NAP angular accelerations were filtered to a CFC600
(1000 Hz) cutoff and numerically integrated and qualitatively and
quantitatively compared to the measured angular velocities from the
angular rate sensors.

Multibody Simulation

An ellipsoid model standing 50th percentile male Hybrid-Ill ATD in
the reference position was struck in the head by an effectively rigid
sphere with a diameter of approximately 7.4 cm using MADYMO (R7.5, TASS
BV, Rijswijk, The Netherlands). Linear and angular kinematics for the
COM of the head were tracked using kinematic time-history outputs.
Linear acceleration outputs were also generated at locations
corresponding to non-COM NAP accelerometer locations. The head was
struck from the front and from the left side using two different
mass-velocity combinations. These two mass-velocity combinations had the
same initial kinetic energies, but the momenta were different. The four
simulations were as follows:

1. The ellipsoid model was struck in the head from the front by a
ball of mass 0.145 kg and a velocity of 26.8 m/s.

2. The ellipsoid model was struck in the head from the front by a
ball of mass 1.308 kg and a velocity of 8.9 m/s.

3. The ellipsoid model was struck in the head from the left side by
a ball of mass 0.145 kg and a velocity of 26.8 m/s.

4. The ellipsoid model was struck in the head from the left side by
a ball of mass 1.308 kg and a velocity of 8.9 m/s.

Simulations were run with a multi-body integration time step of
0.01 ms. Output were written with a time interval equivalent to a sample
rate of 20 kHz per channel.

Linear accelerometer data were filtered to CFC1000. Angular rate
sensor data was analyzed using residual analyses from 1-1650 Hz to
determine a single cutoff frequency that best matched the resultant on a
least-square basis and which best matched the peak value of the
resultant. Differentiation of filtered angular velocity data into
angular acceleration data was performed using two-point differentiation,
Eq 8. Data were also analyzed to determine cutoff frequencies that
yielded least-square or peak value matches within [+ or -]5% of the
optimum value.

RESULTS

ATD Testing

For the six ATD tests, there existed no single cutoff frequency for
the angular rate sensor data that matched NAP-calculated angular
accelerations on both least-squares and peak-matching bases. Only for
test 1 (ATD struck in the face at approximately the level of the COM)
and test 4 (ATD struck near the vertex of the head from the right side)
were the least-square and peak-match optimum cutoff frequency solutions
comparable. An example of least-square versus peak-match cutoff
performance is shown in Figure 3. Cutoff frequencies are described in
Table 1. Maximum resultant linear accelerations and FI[C.sub.15] values
are reported in the Appendix, in Table A1.

The specific equation used for numeric differentiation had less
than a 1% effect on optimum cutoff frequency selection for five of the
six tests, and less than a 3% effect for test 4. The difference in
cutoff frequency selection was smaller for the choice of numeric
differentiation equation than it was for least-square versus peak-match.

By contrast, the selection of cutoff frequency and cutoff frequency
criterion had a large effect on the resulting angular accelerations
(Tables 2a-b). By definition, the match-peak cutoff selection matched
the NAP peak resultant magnitude, and for most tests reasonably matched
the peak magnitude of the primary component axis. The least-square
criterion poorly matched the peak resultant magnitude and the component
axes magnitudes. Most often the result was an underestimation of the
peak resultant magnitude and an overestimation of the component axes
magnitudes. For tests 2 and 4, where resultant estimation was most
accurate, the differences in performance between the least-square and
match-peak criteria were not large. In short, in situations where the
least-square criterion performed best, the match-peak criterion
performed better. This performance was only better on a relative basis,
however; even the best matches yielded component axis errors of almost
100% in magnitude.

The 1000 Hz cutoff frequency generated angular acceleration results
that were essentially unusable. When angular rate sensor data was
low-pass filtered to this cutoff and then numerically differentiated, it
yielded resultant magnitudes that were two to three times larger than
the NAP values and per-axis magnitudes that were an overestimate by two
to three orders of magnitude.

The 300 Hz cutoff was an improvement over the 1000 Hz cutoff, but
resembled a slightly less accurate version of the least-square
criterion. Generally, the 300 Hz cutoff underestimated the peak
resultant by a factor of two or less and overestimated non-primary
component axes by a similar factor. However, the 300 Hz cutoff was very
inaccurate for test 5, where it underestimated the magnitude of the
resultant and of the dominant x-axis by nearly a factor of five.

Neither of the a priori cutoff selections for pre-filtering prior
to numerical differentiation yielded acceptable results. Of the a
posteriori methods, a peak-match criterion was more useful than a
least-square criterion and yielded resultant and primary axis peak
values that were essentially correct. Neither method yielded non-primary
components of the correct magnitude. It is noteworthy that even peak
match badly tracked the time-history of the angular accelerations found
by the NAP, and while it matched the peak values adequately, it did not
necessarily identify those values from the correct impact (Figure 4).
While peak match is the best current method, how to find its cutoffs is
non-trivial and not presently obvious. The Winter method used in prior
studies [27] has not been found to arrive at peak-match cutoffs.

A related question is how one best filters the angular velocity
data measured by an angular rate sensor. SAE J211-2014 [26] recommends
CFC180 (300 Hz) for filtering of linear acceleration or angular velocity
data prior to integration. It also recommends CFC180 for processing of
vehicle control module angular rates. Thus, 300 Hz is a reasonable
starting point for the analysis of filtering of angular velocity data.
It is difficult to determine, without prior knowledge of a solution,
whether a 300 Hz cutoff, a 1000 Hz cutoff, or some other cutoff is most
effective for filtering directly measured angular velocity data (Table
3) In part, this is because drift substantially influences integrated
NAP angular acceleration data. By the end of a 6-second recording, drift
is the primary component of naive calculations. A simple detrend
procedure, which eliminated the average slope of the directly integrated
NAP data was able to recover the angular velocity values with varying
success. More elaborate techniques are feasible, including baselining
and detrending solely in the time window of the signal of interest, but
these make assumptions about the nature of the angular velocity
immediately preceding the impacts, which--for events such as automobile
rollovers or off-dock events--may be non-negligible, unknown, or both.
This difficulty in normalizing integrated angular acceleration data
makes it difficult to provide a reference gold-standard solution by
which to evaluate various selections of filter cutoffs for angular rate
sensor data. In general, a 300 Hz filter of angular rate sensor data
performed most closely to the integration of (de-biased and detrended)
NAP data and exhibited less oscillation about impacts than 1000 Hz data
(Figure 5).

While a 300 Hz cutoff frequency for filtering angular rate sensor
data appeared subjectively better, this selection is not necessarily the
best one. There may exist compelling reasons to select a 1000 Hz cutoff
or some other frequency based on data processing needs, signal frequency
analysis, or adherence to some other standard. A 300 Hz cutoff has
become accepted [26] and commonplace, but at present, this selection
remains arbitrary.

Multibody Simulation

Similarly to the AID tests, for the four multibody simulation
tests, there existed no single cutoff frequency for the angular rate
sensor data that matched noiseless MADYMO output angular accelerations
on both least-squares and peak-matching bases. In a general sense, the
least-square cutoffs were similar to CFC60 and the peak-matching cutoffs
were similar to CFC600. An example of least-square versus peak-match
cutoff performance is shown in Figure 6. Cutoff frequencies are
described in Table 4. As in the ATD tests, the match-peak cutoff
selection matched the MADYMO peak resultant magnitude, and for most
tests reasonably matched the peak magnitude of the primary component
axis. The least-square criterion poorly matched the peak resultant
magnitude and the component axes magnitudes.

There was not a large difference in angular velocity traces
regardless of filter cutoff chosen. The unfiltered NAP array, the NAP
array filtered to CFCIOOO, the ARS with a 1500 rad/s noise amplitude
filtered to 300 Hz, and the ARS with a 1500 rad/s noise amplitude
filtered to CFC1000 performed similarly. Only the ARS with an 18,000
rad/s noise amplitude filtered to 300 Hz noticeably performed poorly. As
noise was defined as a function of sensor range, the ARS array with the
larger range also exhibited larger amounts of noise. For the ARS
systems, sensor range had a larger effect than cutoff frequency. These
are demonstrated in Table 5 and Figure 7.

As in the ATD tests, the selection of cutoff frequency and cutoff
frequency criterion had a large effect on the resulting angular
accelerations (Table 6, Figure 8).

Most often the result was a reasonably accurate estimate of overall
kinematics, but an underestimation of the peak resultant magnitude and
component axes magnitudes. Filtering to CFC1000 often reasonably
estimated the magnitude of peaks, but at significant retained noise and
a loss of correct tracking of the overall kinematics. The data
corresponding to an 18,000 deg/s sensor was substantially noisier than
that corresponding to a 1500 deg/s sensor. It is noteworthy that the
peak resultant angular velocity for test 4--24.4 rad/s resultant and
22.7 rad/s in the x-axis--nearly saturated the 26.2 rad/s range of a
1500 deg/s sensor.

The results of multibody simulation corroborate the ATD testing
finding that there exists no one cutoff frequency which simultaneously
replicates both overall kinematics and peak kinematics. The 300 Hz
filter often used for ARS data yielded accurate angular velocity data
but angular accelerations which underestimated true values. A CFC1000
cutoff frequency also reasonably estimated angular velocities and,
additionally, peak angular acceleration values, but retained significant
noise and poorly tracked non-peak values. It is also noteworthy that the
NAP array reasonably estimated the correct angular kinematics even
without filtering. Maximum resultant linear accelerations and
HI[C.sub.15] values are described in the Appendix, in Table A2.

DISCUSSION

Previous work on the problem of filtering angular velocity data
prior to numeric differentiation into angular acceleration data for
inertial head motions found that a residual analysis technique was
effective in an a priori context - when the solution was not already
known. However, that method assumes that an optimum solution does exist.
The problem encountered in the current study is that at least for some
impact scenarios, an optimum solution may not exist. An a priori
solution method cannot be determined because no effective a posteriori
solution exists. In the six impacts analyzed via ATD, there was no
particular cutoff frequency that was effective for all events. There
also existed no particular cutoff frequency that was effective for
simultaneously preserving peak magnitudes and average signal magnitude.
Even when a cutoff could be found that would preserve the correct value
of the peak resultant magnitude, it did not accurately track the
time-history of the angular acceleration data and in some cases, matched
the correct peak value at the wrong impact event fFigure 4). Thus, at
best, numeric differentiation of angular rate sensor data was only able
to coincidentally match peak values of the angular accelerations as
found by standard NAP techniques, regardless of empirical cutoff
frequency selected.

This effect is not specific or particular to these six tests.
Multibody simulation data with the controlled addition of white noise
exhibits the same problem. This appears to be an inherent limitation to
the application of numerical differentiation techniques in the presence
of white noise in the measurement of short-duration (high-frequency)
impact events. This is depicted in Figure 9, which plots an
"angular velocity" signal consisting of solely white noise
against the angular acceleration derived from it. While the angular
velocity signal has a constant magnitude versus frequency spectrum, the
magnitude of the differentiated angular acceleration signal increases
with increasing frequency. This increasing magnitude as a function of
increasing frequency is a consequence of numeric differentiation in
general, and did not appear to be restricted to any specific equation
used [38]. The net effect is that numeric differentiation techniques
preferentially increase the magnitude of high frequency noise. As shown
in this study, 4-pole phaseless Butterworth filters (as suggested by SAE
J211-2014) are only capable of partially ameliorating this. More exotic
curve-fitting techniques exist for smoothing data prior to
differentiation, but were not analyzed here.

The effect of numeric differentiation is a complicated interaction
between sensor range, sensor frequency response, signal content, noise
content within the system, filter technique, and sample rate. Noise is
increased by increased sensor range and sample rate. Thus, care must be
taken when selecting a sample rate. Butterworth filters have
traditionally been used for digital post-filtering for SAE
J211-2014-compliance [44], as their flat pass-band and rolloff
characteristics accommodated the required corridors. This has most often
taken the form of the phaseless 4th-order filter, with Alem's
correction to account for the effects of the double-filtration inherent
to phaseless filtering on the attenuation at the corner frequency
[44-45]. The 1995 version of J211 specified that post-recording digital
filtration was to be performed only once; while this is no longer
strictly required, filtering is still required to be performed prior to
any non-algebraic operations (such as numeric differentiation or
integration). Thus, in order to accommodate the CFC corridors and to
satisfy the other requirements of SAE J211-2014, the most frequent
solution is a single application of a 4-pole Butterworth filter prior to
numeric operations. Higher-order Butterworth filters will under most
conditions not satisfy the CFC corridors. While non-Butterworth
solutions are allowed, the requirement to match the CFC corridors which
Butterworth filters meet in turn prevents more exotic techniques to
eliminate noise inserted by differentiation. In short, compliance with
SAE J211-2014 means that the numeric differentiation required to convert
angular rate sensor data into angular accelerations will create
problematic noise.

It appears that angular rate sensor technology at present is not
sufficiently noise-free to match NAP performance after numeric
differentiation. This is not to say it is impossible for the state of
the art of angular rate sensors to make up the gap. However, it would
require angular rate sensors to exhibit baseline performance far beyond
that of linear accelerometers. A review of Figure 9 demonstrates that
differentiation of a 10 kHz signal introduces 30 dB of noise between 0
Hz and 1000 Hz. Therefore, an angular rate sensor designed for
differentiation would have to be approximately 30 times less noisy than
a comparable linear accelerometer in order to provide equivalent
performance.

Situations where discrete impacts are separated by non-negligible
amounts of time (more than at most a few tenths of a second) or where
pre-impact kinematics are non-trivial represent worst-case scenarios for
sensors arrays, for they present difficulties for both differentiation
(noise) and integration (drift).

The most straightforward means of avoiding both the numerical
integration step required by the 3-2-2-2 linear array (NAP) and the
numerical differentiation step required by the 3a[omega] array is to
construct a 6a[omega] array, which Kang et al. implemented as a 3-1-1-1
array with the angular rate sensors mounted to each non-centroidal arm
[25]. This implementation yielded angular acceleration results that were
closer to that of the NAP than a 3a[omega] array, transformed peripheral
linear accelerations to the COM better than the 3a[omega] array and
equivalently to the NAP, yielded angular velocity data more accurate
than the NAP, and provided angular velocity data which could be
integrated into angular displacement data at higher accuracy than the
NAP [24-25]. With sufficiently capable angular rate sensors, all
channels of the 6a[omega] array can be filtered to CFC1000, which
preserves time-integrity for transformation purposes [24]. This is ideal
when kinematic measurements are non-centroidal, such as when measuring
living human beings, because data filtered to different cutoff
frequencies are also time-shifted by different amounts, and
transformations performed with mixed cutoff frequencies may yield
temporally-inaccurate kinematics. The primary limitation of
6a[omega]-based arrays is size and channel count. It requires as many
channels and as much physical space as the 3-2-2-2 NAP, and the
individual sensors are often slightly heavier than the linear
accelerometers they replace. This can make the 6a[omega] arrays
difficult to accommodate in applications with limited space, and it is a
more costly solution.

As long as the events can be analyzed as separate components and
long periods of pre-impact or post-impact kinematics are not essential,
a 3-2-2-2 NAP array is an acceptable solution and the manner
traditionally used. Under these conditions, the increased fidelity of a
6a[omega] array may be less valuable than the decreased weight or cost
of a NAP configuration.

Under most conditions, a 3a[omega] array will handle impact events
poorly, due to the inherent noisiness of numerical differentiation and
the present and foreseeable limitations of angular rate sensor
technology. In general, for impacts, both a NAP array and a 6a[omega]
array will provide a more robust solution than a 3a[omega] array. These
should be used unless long-duration analyses are required and impact
events will be of sufficiently low magnitude and low frequency [27].

In this study, physical testing could not be conducted with an ARS
system capable of CFC1000 performance because of the type of ARS sensor
used and in order to maintain consistency between data traces.
Therefore, the linear accelerations had to be filtered to CFC600 for
comparison purposes in this portion accordingly. However, the multibody
simulation results, which were not constrained by instrumentation
availability or capability, demonstrate that a CFC1000-capable ARS
system would not have materially affected any of these conclusions.

SUMMARY/CONCLUSIONS

It was found that there was no specific cutoff frequency that
consistently matched the angular accelerations measured by the 3-2-2-2
array, and the combination of a triaxial linear accelerometer and a
triaxial rate sensor could not simultaneously match the 3-2-2-2 summed
square and peak angular accelerations regardless of cutoff frequency.
The differentiation step to convert angular velocity data into angular
acceleration data was found to insert frequency-dependent noise which
low-pass filtering could not adequately eliminate. While this may be
overcome in the future if the noise floor for angular rate sensors can
be constructed to many times less noise than contemporary linear
accelerometers, this will continue to be a problem inherent to time
differentiation. While angular rate sensors are essential for
long-duration accelerometer recording and are effective for analyses
requiring integration, care must be taken when they are used for impact
analyses. For impact analyses, a 3-2-2-2 array (NAP) or a 6DOF array
paired with an angular rate sensor, such as a 6a[omega] implementation,
will generate results with lower noise, and should be used wherever
possible. In general, for impacts, both a NAP array and a 6a[omega]
array will provide a more robust solution than a 3a[omega] array. These
should be used unless long-duration analyses are required and impact
events will be of sufficiently low magnitude and low frequency.

REFERENCES

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Sebastianelli W.J. Springer: New York, 2014.
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